Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source (Q1298159)

The authors prove the existence of global weak solutions of the problem \(u_t=\Delta u^m-| \nabla u^\alpha| ^q+u^p\) in \(\Omega\times(0,\infty)\), \(u=0\) on \(\partial\Omega\times(0,\infty)\), \(u(x,0)=u_0(x)\geq 0\) in \(\Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\), \(1\leq m<2\alpha\), \(1\leq q<2\), \(1\leq p<\alpha q\) and \(u_0\in L^{m+1}(\Omega)\). The proof is based on the approximation of the problem by nondegenerate problems and on a priori estimates of approximating solutions.